Intertemporal Choice • Persons often receive income in “lumps”; e.g. monthly salary. Chapter Ten • How is a lump of income spread over the following month (saving now for consumption later)? Intertemporal Choice • Or how is consumption financed by borrowing now against income to be received at the end of the month? Present and Future Values Future Value • Begin with some simple financial • E.g., if r = 0.1 then $100 saved at the start arithmetic. of period 1 becomes $110 at the start of • Take just two periods; 1 and 2. period 2. • Let r denote the interest rate per period. • The value next period of $1 saved now is the future value of that dollar. 1 Future Value Present Value • Suppose you can pay now to obtain $1 • Given an interest rate r the future value at the start of next period. one period from now of $1 is • What is the most you should pay? FV === 1 +++ r. • $1? • No. If you kept your $1 now and saved • Given an interest rate r the future value it then at the start of next period you one period from now of $m is would have $(1+r) > $1, so paying $1 FV === m(1 +++ r). now for $1 next period is a bad deal. Present Value Present Value • Q: How much money would have to be • The present value of $1 available at the saved now, in the present, to obtain $1 at start of the next period is the start of the next period? 1 PV === . • A: $m saved now becomes $m(1+r) at 1 +++ r the start of next period, so we want the • And the present value of $m available at value of m for which the start of the next period is m(1+r) = 1 m PV === . That is, m = 1/(1+r), +++ the present-value of $1 obtained at the 1 r start of next period. 2 The Intertemporal Choice Present Value Problem • E.g., if r = 0.1 then the most you should pay now for $1 available next period is • Let m 1 and m 2 be incomes received in 1 periods 1 and 2. PV === ===$0⋅⋅⋅ 91 . • Let c and c be consumptions in periods 1 1 +++ 0 ⋅⋅⋅ 1 1 2 • And if r = 0.2 then the most you should and 2. pay now for $1 available next period is • Let p 1 and p 2 be the prices of consumption in periods 1 and 2. 1 PV === ===$0⋅⋅⋅ 83 . 1 +++ 0 ⋅⋅⋅ 2 The Intertemporal Choice Problem The Intertemporal Budget • The intertemporal choice problem: Constraint Given incomes m 1 and m 2, and given • To start, let’s ignore price effects by consumption prices p 1 and p 2, what is the supposing that most preferred intertemporal consumption bundle (c 1, c 2)? p1 = p 2 = $1. • For an answer we need to know: – the intertemporal budget constraint – intertemporal consumption preferences. 3 The Intertemporal Budget The Intertemporal Budget Constraint Constraint c2 • Suppose that the consumer chooses not So (c 1, c 2) = (m 1, m 2) is the to save or to borrow. consumption bundle if the consumer chooses neither to • Q: What will be consumed in period 1? save nor to borrow. • A: c 1 = m 1. • Q: What will be consumed in period 2? m2 • A: c 2 = m 2. 0 0 m1 c1 The Intertemporal Budget The Intertemporal Budget Constraint Constraint • Now suppose that the consumer spends • Period 2 income is m 2. nothing on consumption in period 1; that is, • Savings plus interest from period 1 sum c1 = 0 and the consumer saves to (1 + r )m 1. s1 = m 1. • So total income available in period 2 is • The interest rate is r. m2 + (1 + r )m 1. • What now will be period 2’s consumption • So period 2 consumption expenditure is level? = + + c2 == m2 ++ (1 ++ r )m1 4 The Intertemporal Budget The Intertemporal Budget Constraint Constraint c2 c2 = + + the future-value of the income (c1,c2 ) == (((0,m2 ++ (1 ++ r)m1))) m +++ m +++ 2 endowment 2 is the consumption bundle when all +++ +++ ()1 r m1 ()1 r m1 period 1 income is saved. m2 m2 0 0 0 m1 c1 0 m1 c1 The Intertemporal Budget The Intertemporal Budget Constraint Constraint • Now suppose that the consumer spends • Only $m 2 will be available in period 2 to everything possible on consumption in pay back $b 1 borrowed in period 1. period 1, so c = 0. 2 • So b 1(1 + r ) = m 2. • What is the most that the consumer can • That is, b 1 = m 2 / (1 + r ). borrow in period 1 against her period 2 • So the largest possible period 1 income of $m ? 2 consumption level is • Let b denote the amount borrowed in 1 m2 period 1. c=== m +++ 1 1 1 +++ r 5 The Intertemporal Budget The Intertemporal Budget Constraint Constraint c2 = + + c2 = + + (c1,c2 ) == (((0,m2 ++ (1 ++ r)m1))) (c1,c2 ) == (((0,m2 ++ (1 ++ r)m1))) m +++ m +++ 2 is the consumption bundle when all 2 is the consumption bundle when +++ +++ ()1 r m1 period 1 income is saved. ()1 r m1 period 1 saving is as large as possible. m (,),c c=== m +++ 2 0 1 2 1 +++ the present-value of 1 r m m is the consumption bundle 2 the income endowment 2 when period 1 borrowing is as big as possible. 0 0 m m c m m c 0 1 m +++ 2 1 0 1 m +++ 2 1 1 1 +++ r 1 1 +++ r The Intertemporal Budget The Intertemporal Budget Constraint Constraint c2 = + + (c1,c2 ) == (((0,m2 ++ (1 ++ r)m1))) • Suppose that c units are consumed in m +++ 1 2 is the consumption bundle when +++ period 1. This costs $c 1 and leaves m 1- c1 ()1 r m1 period 1 saving is as large as possible. saved. Period 2 consumption will then be = + m2 (,),c1 c 2== m 1 ++ 0 = + + − 1 +++ r which is c2 == m2 ++ (1 ++ r)( m1 −− c1) m2 is the consumption bundle === −−− +++ +++ +++ +++ when period 1 borrowing c2 (1 r)c1 m2 (1 r )m1. is as big as possible. 0 m m c 0 1 m +++ 2 1 slope intercept 1 1 +++ r 6 The Intertemporal Budget The Intertemporal Budget Constraint Constraint c2 c === −−−(1 +++ r)c +++ m +++ (1 +++ r )m . (1 +++ r )c +++ c === (1 +++ r )m +++ m + 2 1 2 1 1 2 1 2 m2 ++ is the “future-valued” form of the budget (1 +++ r)m S slope = -(1+r) 1 a constraint since all terms are in period 2 vi ng values. This is equivalent to c m B c +++ 2 ===m +++ 2 o 1 +++ 1 +++ m2 rr 1 r 1 r ow i which is the “present-valued” form of the ng 0 constraint since all terms are in period 1 m m c 0 1 m +++ 2 1 values. 1 1 +++ r The Intertemporal Budget Intertemporal Choice Constraint • Now let’s add prices p 1 and p 2 for • Given her endowment (m 1,m 2) and prices consumption in periods 1 and 2. p1, p 2 what intertemporal consumption • How does this affect the budget constraint? bundle (c 1*,c 2*) will be chosen by the consumer? • Maximum possible expenditure in period 2 is + + m2 ++ (1 ++ r )m1 so maximum possible consumption in period 2 is +++ +++ = m2(1 r )m 1 c 2 == . p2 7 Intertemporal Choice Intertemporal Choice • Similarly, maximum possible expenditure • Finally, if c 1 units are consumed in period in period 1 is 1 then the consumer spends p 1c1 in period m 1, leaving m - p c saved for period 1. m +++ 2 1 1 1 1 1 +++ r Available income in period 2 will then be so maximum possible consumption in period 1 is so m +++ (1 +++ r)( m −−− p c ) +++ +++ 2 1 1 1 === m1m 2 / (1 r ) c1 . p c === m +++ (1 +++ r)( m −−− p c ). p1 2 2 2 1 1 1 The Intertemporal Budget Intertemporal Choice Constraint = + + − c2 (1 +++ r )p c +++ p c === (1 +++ r )m +++ m p2c2 == m2 ++ (1 ++ r )( m1 −− p1c1 ) + + 1 1 2 2 1 2 (1 ++ r )m1++ m 2 rearranged is p2 + + = + + S −−−+++ p1 (1 ++ r )p1c1 ++ p2c 2 == (1 ++ r )m1 ++ m 2 . a Slope = ()1 r vi p This is the “future-valued” form of the ng 2 budget constraint since all terms are B o expressed in period 2 values. Equivalent m2/p 2 rr ow to it is the “present-valued” form i ng + p 2 =+ m 2 p1 c 1 ++ c2== m 1 ++ 0 1 +++ r 1 +++ r 0 m1/p 1 c1 + + where all terms are expressed in period 1 m1++ m 2 / (1 ++ r ) values. p1 8 Price Inflation Price Inflation • Define the inflation rate by π where • We lose nothing by setting p 1=1 so that p = 1+ π . p (1 +++ πππ ) === p . 2 1 2 • Then we can rewrite the budget constraint • For example, p m π = 0.2 means 20% inflation, and p c +++ 2 c=== m +++ 2 1 1 +++ 2 1 +++ π = 1.0 means 100% inflation. as 1 r 1 r 1 +++ πππ m c +++ c=== m +++ 2 11 +++ r 2 1 1 +++ r Price Inflation Price Inflation +++ πππ • When there was no price inflation + 1 =+ m 2 c1++ c 2== m 1 ++ 1 +++ r 1 +++ r (p 1=p 2=1) the slope of the budget rearranges to constraint was -(1+r). 1 +++ r m • Now, with price inflation, the slope of c === −−− c +++ 1( +++ πππ) 1 +++ m π 2 1 +++ πππ 1 1 +++ r 2 the budget constraint is -(1+r)/(1+ ).